The Quarterly Projection Model - CERGE-EI

The Quarterly Projection Model

Franti²ek BrázdikMacroeconomic Forecasting Division

[email protected]

Czech National Bank

November 2011

Czech National Bank QPM 1 / 77

Outline

1 Trend and cycles

2 Structure of the Quarterly Projection Model

3 Parameters setup

4 Properties of the Model


Trend and cycles

Outline

1 Trend and cycles


3 Parameters setup



Trend and cycles

Time series analysis

Analysis of time series data is based on smoothing past data inorder to separate the underlying pattern in the data series fromrandomness.

The underlying pattern then can be projected into the futureand used as the forecast.

The underlying pattern can also be broken down into subpatterns to identify the component factors that in�uence each ofthe values in a series: decomposition

Decomposition methods: identify separate components of thebasic underlying pattern that tend to characterize economics andbusiness series.


Trend and cycles

In search for trends


Trend and cycles

Decomposition Techniques

Goal: separation of data into several unobservable components,generally in an additive or multiplicative form.

Components: trend, seasonal pattern, cycle, and residual orirregular pattern

Seasonal component: the periodic �uctuations of constantlength

Trend-cycle component: long term changes in the level of series


Trend and cycles Detrending methods

Detrending

Trend Component: The tendency of a variable to grow overtime, either positively or negatively.

Basic forces in trend: population change, price change,technological change, productivity change, product life cycles

The long term movements or trend in a series can be describedby a straight line or a smooth curve.

The long-term trend is estimated from the seasonally adjusteddata for the variable of interest

Interpretation:I Trends: long run equilibriumI Gaps: cyclical �uctuations



Trend analysis

Assume seasonally adjusted dataTrend-Cycle decomposition: Series = Trend + Cycle + NoiseNo general-automatic techniques for detrendingSimple techniques: Smoothing

I Moving average: The average eliminate some higher frequencynoise in the data, and leaves a smooth trend-cycle component.What order to use?

I Simple centered moving average: can be de�ned for any oddorder. A moving average of order k, is de�ned as the averageconsisting of an observation and the m = (k-1)/2 points oneither side.

I Centered moving average: take the simple centered movingaverage, assign weights and create weighted average

Advanced techniques of detrending:I Fitting a polynomialI Using a structural model



Detrending techniques overview I

Watson detrending: greater business cycle persistence; trendcomponent follows a random walk with drift and cyclicalcomponent is a stationary �nite order AR process.

Harvey-Clark detrending: local linear trend model

Hodrick-Prescott �lter: univariate method

Kalman �lter: multivariate method, structural method

Bandpass �lter: not widely used, frequency domain analysis



Detrending techniques overview II

Detrending comparison: US GDP gap


QPM structure

Outline

1 Trend and cycles


3 Parameters setup



QPM structure

Motivation for QPM

Separate econometric methods: Inconsistencies

Short experience with FPAS: Forecasting and Policy AnalysisSystem

State:I Insu�cient data and experience to support advanced modelI Need to increase participation of other departments and bankboard

I Communication of results: support for decisionI Need for research tool

The further step on the way to complex structural models:DSGE


QPM structure

Features of QPM

Re�ects in�ation targeting regime:I In December 1997: after an exchange rate crisisI CNB adopted a series of end-year in�ation targetsI Regime proved very e�ective in combating in�ation andanchoring

I Evolution toward a more transparent in�ation targeting regimewhere monetary policy is anchored by a medium-termperspective

I Change to point in�ation target: In�ation target bandI The character of the regime was further enhanced by publicationof unconditional forecasts

Linked to quarterly data

Small open-economy gap model


QPM structure

Model of trends and cycle

Two separate blocks:I Long run equilibrium trendsI Cyclical �uctuations - gapsI These blocks are separableI Super-neutrality: no long-run trade-o� between output andin�ation

85 equations at start

Further extensions


QPM structure QPM trends

Long Run Trends

First step: �lter trend seriesI History - estimated by a simple statistical model (Kalman �lter)and expert judgement

I Forecast - exogenous (expert judgement), respecting steadystate properties of QPM

Important equilibrium values:I Real output growthI Real wage growthI Real exchange rate appreciationI Real interest rateI Stationarity is required: growth rates in focus

Monetary decisions have small impact on long term real trends


QPM structure QPM cycles

Cyclical Part of QPM

Description of the position of the Czech economy

Monetary policy characteristics:I In�ation targeting regimeI Forward looking policyI Focus on deviations from the target −→ reaction to expectedin�ation a year ahead

I Floating exchange rate - endogenous

Description of behavior economic agents includes forwardlooking components

Price frictions:I Wage stickinessI Final price stickinessI Expectation stickiness


QPM structure QPM scheme

Scheme of model



Real Economy I

IS curve (Aggregate demand):

Output: function of lagged output, the real interest rate, thereal exchange rate and foreign demand and interest rate

Includes impact of a change in interest rates with longer maturityon aggregate demand and take into account expectations aboutyield-curve on the dynamic properties of the model

Real impact of monetary policy in a sticky-price model of a smallopen economy

Marginal costs: cost of producing additional unit of a good



Real Economy II

Real Marginal Costs Gap:

Approximation of in�ationary pressures from the real economy.

Marginal costs consist of the costs arising from the increasingvolume of production (the "output gap") and wage costs (the"real wage gap").

A positive real marginal cost gap implies an in�ationary e�ect ofthe real economy

mct = λyt + wrt



Real Economy III

Output Gap:

Standard economic theory: higher real interest rate reduceaggregate demand by increasing the reward to saving

Output gap: responds negatively to the di�erence between thereal interest rate and its equilibrium value

Open economy: the exchange rate matters

Currency appreciation will, all else equal, make domestic goodsmore expensive in foreign markets and reduce demand fordomestic goods abroad; cheaper imports may displace domesticgoods

yt = α1yt−1 − rmcit−1 + α2yft + εyt

rmcit = β1

(β3rct + β4r4t + (1− β3 − β4) r4

f

t

)+ β2zt



Real Economy IV

Real Wage Gap:

Introduced in January 2007

Wage costs are above their equilibrium level, they have anin�ationary e�ect

The e�ect of a deviation of the current level of the average realwage from its equilibrium level, which in the long run rises at thesame rate as equilibrium real output (non-accelerating in�ationreal output)

wrt = wrt−1 +wt

4− πt

4− 4wrt

4+ εwr

t



Real Economy V

Unemployment:

Okun law

Unemployment gap depends on its lag output gap.



Phillips Curves I

Price In�ation:

Standard Phillips curve has been modi�ed for a small openeconomy

Blocks for various goods

Import price e�ects

Wage setters derive their nominal wage demand real consumerwage

x for fuel, food, or adjusted excl. fuel in�ation

Administered prices are exogenous in baseline



Phillips Curves II

πxt

= γx1

(π4Mx

t+44z

xt

)+ γx

2

(Eπ4t +44z

xt−44zt

)+(1− γx

1− γx

2

)πxt−1

+ γx3mct + επ

x

t

Wage In�ation:

Erceg, C.J., Henderson, D.W., Levin A.T.: Optimal monetarypolicy with staggered wage and price contracts, 2000

wt = δ1Ew4t + (1− δ1)wt−1 − δ2(wrt − δ3yt

)+ εw

t



Expectations I

Price In�ation Expectations:

Expected in�ation: a weighted combination of abackward-looking and a forward-looking component (theexpected value of overall CPI in�ation over the next fourquarters)

Overall CPI: an explicit link between changes in administeredand energy prices and pressures on the rate of in�ation formarket prices

Eπ4t = λ1πt+1 +(1− λ1

)πt−1 + εE4

t

Wage In�ation Expectations:

Ew4t = λ2wt+1 +(1− λ2

)wt−1 + εEw4

t



Uncovered interest rate parity

Nominal Exchange Rate:

UIP condition: arbitrage condition; international investors willequalize e�ective rates of return on investments in di�erentcurrencies, allowing for any country-speci�c risk premiums

foreign investor expecting a depreciation (appreciation) of thekoruna will demand a higher (lower) return from Czech assets

Moving average form

st = φst+1 + (1− φ)

(st−1 + 2

(Etπ

4− Etπ

f

4

)+ 24 zt

)

+it

4− i

ft

4− premt + εs

t



Reaction Function

Nominal Interest Rate:

Forward-looking reaction function

CPI in�ation expected to be above the target rate: central bankpush up the short-term

Excess demand: the central bank increases short-term interestrate

Long-term level for rates and some additional dynamic structure

Interest rate inertia: interest rate smoothing

it = ψit−1 + (1− ψ)(ineutralt

+ Πt

)+ εi

t

ineutralt

= rt + π4t+4 + εit

Πt = κ1

(π4t+4 − π4targett+4

)+ κ2yt


Parameters

Outline

1 Trend and cycles


3 Parameters setup



Parameters

Calibration vs. Estimation

QPM is calibrated, partially estimated

Problems in estimation:I Short data sampleI Structural changes in economyI Changes of monetary policy regimeI It is impossible to estimate some parameters: identi�cationproblems


Parameters

Calibration of QPM

Parameters setup:

Restrictions on parameters originating from economic theory

Parameters are set to mach the properties of data

Responses to structural shocks

Parameters checks:

Reactions to shocks

Residuals

In-sample simulations

Curve-�tting estimates


Model Properties

Outline

1 Trend and cycles


3 Parameters setup



Model Properties

Price shock I

Positive shock to the output gap

Upward pressure on in�ation

Currency depreciation

Central bank increases interest rate

Cumulative e�ect on output is very close to zero: feature oflinear models;

O�setting of excess supply to counteract the e�ects of shocksthat create excess demand


Model Properties

Price shock II


Model Properties

Aggregate demand shock I

Positive shock to the output gap

Upward pressure on in�ation

Currency depreciation

Central bank increases interest rate

Cumulative e�ect on output is very close to zero: feature oflinear models;

O�setting of excess supply to counteract the e�ects of shocksthat create excess demand


Model Properties

Aggregate demand shock II


Model Properties

Exchange rate shock I

Depreciation acts to increase aggregate demand, opening apositive output gap


Model Properties

Exchange rate shock II


Model Properties

In�ation target change I

Lower the target rate of in�ation by one percentage point

To achieve disin�ation: raise the short rate

Appreciation: Import prices fall

The combined e�ect of the import price decline and the excesssupply gap works to gradually pull down the rate of in�ation

Note: purely nominal shock, and since the model issuper-neutral, there is no change to any real equilibrium in thisshock, including the real exchange rate. The nominal exchangerate changes, of course, with the cumulative

Cumulative e�ects on output and employment

Sacri�ce ratio: a cumulative loss of output vs. lower in�ation bya percentage point


Model Properties

In�ation target change II


Model Properties Data �tting

Residuals I

Con�ict between estimated parameters and calibrated

The parameters have to be chosen so as to give reasonablemodel behavior

Examined how well the model performs over the historical sample

Identify systematic biases



Residuals II



In-Sample Simulations: CPI



In-Sample Simulations: Ex. rate



In-Sample Simulations: GDP



In-Sample Simulations



Modeling tools

Implementation in Matlab

IRIS by Jaromír Bene²




Appendix Filters

Univariate �ltering I

Hodrick-Prescott �lter: optimally extracts a trend which isstochastic but moves smoothly over time and is uncorrelatedwith the cyclical component

Mathematics of HP �lter:I Decomposition: yt = τt + ctI Solve:min

∑T

t=1(yt − τt)

2 + λ ∗∑

T−1

t=2[(τt+1 − τt)− (τt − τt−1)]

2

I λ = 100 ∗ (number of periods in a year)2

Assumption that the trend is smooth is imposed by assumingthat the sum of squares of the second di�erences of τt is small

Sensitivity of the trend to short-term �uctuations is achieved bymodifying a multiplier λ


Appendix Filters

Univariate �ltering II

Drawbacks:

I One-time permanent shock, split growth rates present: Filteridenti�es non-existing shifts in the trend

I It pushes noise in data to Normal distributionI Misleading predictive outcome: Analysis is purely historical andstatic


Appendix Filters

Univariate �ltering III

Trend:


Appendix Filters

Univariate �ltering IV

Gap:


Appendix Kalman �lter

Kalman �lter I

Separate the cyclical component of a time series from raw data

Can handle more series and exploit relations between them

Kalman �lter is a powerful tool for:I EstimationI PredictionI Smoothing

Kalman �lter:I Online estimation procedureI States are estimated, when the new observations are coming in

Kalman smoother:I O�-line estimation procedureI The state estimation of is not only based on all previousobservations, but also on all later observations



Kalman �lter II

F is the statetransition model

B is the control-inputmodel

H is the observationmodel

w is the process noise

z is the measurement

v is the measurementerror

u is the exogenouscontrol



Kalman �lter structure


Appendix Simple �ltering model

Description of variables

Measurement variables: ∆EU LGDP ,EU LGDPGAP EXPERT

State variables: ∆EU LGDP EQ,MU,EU LGDPGAP

Exogenous-variables: EU RMCIGAP

Shocks: ν's

Coe�cients: a1, a2, a3 and µSS

Variance: σ1, σ2, σ3, σ4

Remark: In the following slides the �ltering is actually smoothing


Appendix Simple �ltering model

Description of model

Measurement equations:

∆EU LGDP = ∆EU LGDP EQ +

+ 4 ∗ (EU LGDPGAP − EU LGDPGAP{−1})EU LGDPGAP = EU LGDPGAP EXPERT + σ4 ∗ ν4

State equations:

∆EU LGDP EQ = µ + σ1 ∗ ν1µ = (1− a3) ∗ µSS + a3 ∗ µ{−1}+ σ3 ∗ ν3

EU LGDPGAP = a1 ∗ EU LGDPGAP{−1}+

+ a2 ∗ EU RMCIGAP{−1}+ σ2 ∗ ν2


Appendix Filtering results

Filtering results: EU Eq. trajectories



Filtering results: EU Gap estimate



Filtering results: Removing volatility



Model setting: Changes in volatility of gap σ2


Appendix Complex model

Filtering domestic variables

First step:I Decompose real variables: trend and cycleI Simple model for: Real interest rate, Real exchange rate,Exchange risk premium

Second step:I Utilize measurement of in�ation and wage growthI Fit simple backward-looking Phillips curves: relation betweenin�ation and output gap

I Fit IS curve: relation between output gap and gaps in realinterest and exchange rate

I Decompose: domestic output, real wage, unemployment



Filtering results: Domestic Eq. trajectory



Filtering results: Domestic output gap


Appendix Expert judgement

Description: Second step model

Measurement variables: DOT LGDP,DOT UNR,PIE CORE ,PIE W ,DOT LWR, LWR GAP EXPERT , LGDP GAP EXPERT ,UNR GAP EXPERT

State variables: DOT LGDP EQ,MU, LGDP GAP,DOT UNR EQ,UNR GAP,PIE CORE S,PIE W S,DOT LWR EQ, LWR GAP

Exogenous-variables:RRC GAP,RR4 GAP,EU RR4 GAP, LZ GAP,EU LGDP GAP,PIE M XENERGY 4,DOT LZ CORE EQ4,DOT LZ EQ4,E0 CORE4,E0 PIE W4, DOT LWR PRIOR,E0 PIE4

Shocks: νs

Variance: σs


Appendix Model details

Model I

Measurement equations:

DOT LGDP = DOT LGDP EQ + 4 ∗ (LGDP GAP − LGDP GAP{−1})DOT UNR = DOT UNR EQ − 4 ∗ (UNR GAP − UNR GAP{−1})PIE CORE = PIE CORE S

PIE W = PIE W S

DOT LWR = DOT LWR EQ + 4 ∗ (LWR GAP − LWR GAP{−1})LWR GAP = LWR GAP EXPERT + std w3 ∗ ν LWR GAP EXPERT

LGDP GAP = LGDP GAP EXPERT + std w1 ∗ ν LGDP GAP EXPERT

UNR GAP = UNR GAP EXPERT + std w2 ∗ ν UNR GAP EXPERT


Appendix Model details

Model II

State equations:

DOT LGDP EQ = MU{−1} + a1 ∗ DOT UNR EQ + std v1 ∗ ν DOT LGDP EQ

LGDP GAP = LGDP GAP C01 ∗ LGDP GAP{−1} − RMCI GAP C02 ∗ (b2 ∗ RRC GAP{−1}+ b3 ∗ RR4 GAP{−1} + b4 ∗ EU RR4 GAP{−1})

−RMCI GAP C01 ∗ LZ GAP{−1} +LGDP GAP C02 ∗ EU LGDP GAP + std v2 ∗ ν LGDP GAP

MU = (1− a3) ∗MU SS + a3 ∗MU{−1} + std v3 ∗ ν MUDOT UNR EQ = std v4 ∗ ν DOT UNR EQ

UNR GAP = UNR GAP C01 ∗ UNR GAP{−1}+ UNR GAP C02 ∗ LGDP GAP + std v5 ∗ ν UNR GAP

PIE CORE S = PIE CORE C01 ∗ (PIE M XENERGY 4 + DOT LZ CORE EQ4)

+ PIE CORE C02 ∗ (PIE CORE C05 ∗ E0 CORE4+ (1− PIE CORE C05) ∗ E0 PIE4)+ (1− PIE CORE C01− PIE CORE C02) ∗ PIE CORE S{−1}+ RMC GAP C01 ∗ PIE CORE C03 ∗ LGDP GAP

+ PIE CORE C03 ∗ LWR GAP

+ std v6 ∗ ν PIE CORE

PIE W S = PIE W C01 ∗ E0 PIE W4 + (1− PIE W C01) ∗ PIE W S{−1}+ PIE W C02 ∗ (LWR GAP − PIE W C03 ∗ LGDP GAP) + std v7 ∗ ν PIE W

DOT LWR EQ = DOT LGDP EQ + DOT LWR PRIOR + std v8 ∗ ν DOT LWR EQ

LWR GAP = f 1 ∗ LWR GAP{−1} + std v9 ∗ ν LWR GAP

Fixing: unemployment gap in 1999Czech National Bank QPM 66 / 77

Appendix Expert judgement simulations

Filtering results: Expert judgement


Appendix Expert judgement simulations

Filtering results: Expert judgement


Appendix Advanced �ltering

Advanced �ltering

Criticism of simple models: lack of reference to unemployment

J. Galí,F. Smets and R. Wouters (2011):I Address this issue in an extended modelI Conclusion: Model-based output gap resembles conventionalmeasures of the cyclical component of log GDP.

I Comparison of a variety of statistical detrending methodsI HP �lter, band-pass �lter, quadratic detrending, and theCongressional Budget O�ce's measure



Advanced �ltering



In search for future trends



List of Variables I

a gap of the variable aa trend (equilibrium) value of the variable aaf variable a for the foreign countryεa residual in the equation for the variable a

mc real marginal costsy real outputrw real wagermci real monetary condition indexr4 real 1Y interbank rater real 3M interbank raterc real rate of newly-issued bank loansz real exchange rate



List of Variables II

π4target in�ation target (y-o-y)π price in�ation (q-o-q)π4 price in�ation (y-o-y)w wage in�ation (q-o-q)w4 wage in�ation (y-o-y)π4M imported in�ation (y-o-y)s nominal exchange rateprem risk premiumi nominal short-term interest rateineutral policy neutral short-term interest rate

α, β, γ, δ, φ, ψ, κ, λ parameters


Appendix Literature

For Further Reading I

Cbo'S Method For Estimating Potential Output: An Update,http://www.cbo.gov/doc.cfm?index=3020&type=0

Jordi Galí and Frank Smets and Rafael WoutersUnemployment In An Estimated New Keynesian Model,National Bureau Of Economic Research,vol. 17084, 2011

Peter K. ClarkThe Cyclical Component of U.S. Economic Activity,The Quarterly Journal of Economics,vol. 102,1987


Appendix Literature

For Further Reading II

Rudolph E. KalmanA New Approach to Linear Filtering and Prediction ProblemsTransactions of the ASME�Journal of Basic Engineering, vol. 82,Series D, 1960

Greg Welch and Gary BishopAn introduction to the Kalman �lter.University of North Carolina, July, 2006; 2000.

Harvey, Andrew C, 1985Trends and Cycles in Macroeconomic Time SeriesJournal of Business and Economic Statistics, Vol. 3 p. 216


Appendix Literature

For Further Reading III

Watson, Mark M, 1986Univariate Detrending Methods with Stochastic TrendsJournal of Monetary Economics, Vol. 18, p. 49

Athanasios Orphanides and Simon van Norden, 2002The Unreliability of Output-Gap Estimates in Real TimeThe Review of Economics and Statistics, Vol. 84, Num. 4


Appendix Literature

Additional one ...


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The Quarterly Projection Model - CERGE-EI